Question

Consider a function $f\left(x\right)=x^x,\forall x\in \left[1,\in fty\right)$ . If $g\left(x\right)$ is the inverse function of $f\left(x\right)$ , then the value of $g^{{}^{\prime }}\left(4\right)$ is equal to

• A. $lo{g}_{2}e$
• B. $\frac{1}{2}lo{g}_{2e}e$
• C. $\frac{1}{4}lo{g}_{2e}e$
• D. $\frac{1}{2}lo{g}_{e}2e$

Answer 1

Answer: (C)

$f\left(x\right)=x^x\Rightarrow logf\left(x\right)=xlogx$
Differentiating with respect to $x$ , we get,
$f^{{}^{\prime }}\left(x\right)=x^x\left(1+logx\right)$
Now, by property $g^{\prime }(f(x))=\frac{1}{f^{\prime }(x)}$ { $\because g\left(x\right)$ is the inverse of $f\left(x\right)$ }
Putting $x=2$
$g^{{}^{\prime }}\left(f\left(2\right)=\frac{1}{f^{{}^{\prime }}\left(2\right)}\Rightarrow g^{{}^{\prime }}\left(4\right)=\frac{1}{2^2\left(1+log2\right)}=\frac{1}{4}{\left(log\right)}_{2e}e\right)$